Contact Mechanics and Elements of Tribology Foreword

Vladislav A. Yastrebov

MINES ParisTech, PSL Research University, Centre des Matériaux, CNRS UMR 7633, Evry, France

@ Centre des Matériaux February 8, 2016 Outline

Acquaintance Questionnaire Course content Historical remark Complexity Notations Questionnaire

Please fill in the questionnaire Message to DMS students :

Rank all suggested topics1 for your report in the order of decreasing attractiveness Before the lunch one topic will be assigned to everybody On Friday (12/02) you will have 10 minutes to present briefly your topic to other students If you are not a DMS student but would like to follow, you are welcome Basava, Andrei and Paolo will also present rapidly their research projects on 12/02

By Friday (19/02) you will need to send me your report on your topic ( 10 pages in Français/English) ≈ Criteria of evaluation : no plagiarism, scientifically sound report

1or suggest a different one but not related to your master project

V.A. Yastrebov 3/10 Course content Lectures : 1 Motivation : industrial applications 2 Contact Mechanics I & II 3 Contact at small scales : surface roughness 4 Computational Contact Mechanics 5 Contact rheology & friction laws 6 Elements of tribology : friction, adhesion, wear 7 Wear and fretting (by Henry Proudhon) 8 Multiphysical problems in contact 9 Your presentations

V.A. Yastrebov 4/10 Course content Lectures : 1 Motivation : industrial applications 2 Contact Mechanics I & II 3 Contact at small scales : surface roughness 4 Computational Contact Mechanics 5 Contact rheology & friction laws 6 Elements of tribology : friction, adhesion, wear 7 Wear and fretting (by Henry Proudhon) 8 Multiphysical problems in contact 9 Your presentations Practice : 1 Chalk and blackboard : Boussinesq, Cerruti, Flamant 2 Chalk and blackboard : Hertz contact 3 Computer : rough contact 4 Computer : frictionless and frictional cylindrical contact 5 Computer : elasto-plastic spherical indentation

V.A. Yastrebov 5/10 Historical remark

Use of friction and overcoming of friction Frictional heat - lighting of fire > 40 000 years ago. Ancient Egypt - lubrication of surfaces with oil > 5 000 years ago.

V.A. Yastrebov 6/10 Historical remark

First scientific studies on contact and friction

Leonardo da Vinci [1452-1519] first friction laws and many other tribological topics

Isaac Newton [1687] From Leonardo da Vinci’s notebook Newton’s third law for bodies interaction

Guillaume Amontons [1699] rediscovered friction laws

Leonhard Euler [1707-1783] roughness theory of friction

Roughness theory of friction

V.A. Yastrebov 7/10 Historical remark

First scientific studies on contact and friction

Charles-Augustin de Coulomb [1789] friction independence on sliding velocity and roughness, the influence of the time of repose

Heinrich Hertz [1881-1882] Photoelasticity analysis of Hertz contact problem the first study on contact of (fringes correspond to max shear stresses) deformable solids

Holm [1938], Ernst and Merchant [1940], Bowden and Tabor [1942] difference between apparent and real contact areas, adhesion theory.

Apparent and real areas of contact

V.A. Yastrebov 8/10 Complexity

Zoom on the contact interface Between different heterogeneous materials with specific rough surfaces Under various loads

V.A. Yastrebov 9/10 Involved physical phenomena

Primary phenomena (atomic / electronic interactions) 1 Contact (non-penetration) 2 Friction (tangential resistance) 3 Adhesion (resistance to tension) 4 Surface energy 5 Quantum effects (tunneling, Casimir effect) Secondary phenomena 1 Wear 2 Lubrication (fluid-structure interaction) 3 Environmental effects (oxidation) 4 Aging, diffusion 5 Heat production 6 Heat and electricity transfer

V.A. Yastrebov 10/10 Contact complexity : physics and mathematics

Particular difficulties related to contact problems : multiphysical aspects, mathematical aspects

Fractality of surfaces Interface chemistry Hardly accessible contact interface Generation and diffusion of heat Multiscale and multiphysical nature of friction

V.A. Yastrebov 11/10 Contact complexity : physics and mathematics

Particular difficulties related to contact problems : multiphysical aspects, mathematical aspects

One of the most hard problems in mechanics Lack of standard optimization problem Non-convexity and non-differentiability Non-continuous character Bad scalability

V.A. Yastrebov 12/10 Notations

Vectors and tensors a, α scalars a b = c scalar (dot) product • • · b vectors a b = c vector (cross) product • • × C, β 2nd order tensors a b = C tensor product • = = • ⊗ = 4 D 4th order tensors AT transposition • = • = a = B gradient operator a = c divergence operator •∇ = •∇· a = B rotor operator I = e e unitary 2nd order tensor •∇× = •= i ⊗ i Mechanics σ Cauchy stress tensor ε Small strain tensor • = • = g, gn gap, normal gap ξ position vector in parent space • •  penalty parameter n∼ outward unit normal vector • • ∂% ∂% λ, λn, λt lagrange multipliers , surface tangent vectors • • ∂ξ1 ∂ξ2 σn =(σ n) n contact pressure f Coefficient of friction • =· · • V.A. Yastrebov 13/10 Welcome to CMET course ! ,